10 14 16 triangle

Acute scalene triangle.

Sides: a = 10   b = 14   c = 16

Area: T = 69.28220323028
Perimeter: p = 40
Semiperimeter: s = 20

Angle ∠ A = α = 38.21332107017° = 38°12'48″ = 0.66769463445 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 81.78767892983° = 81°47'12″ = 1.42774487579 rad

Height: ha = 13.85664064606
Height: hb = 9.89774331861
Height: hc = 8.66602540378

Median: ma = 14.17774468788
Median: mb = 11.35878166916
Median: mc = 9.16551513899

Inradius: r = 3.46441016151
Circumradius: R = 8.08329037687

Vertex coordinates: A[16; 0] B[0; 0] C[5; 8.66602540378]
Centroid: CG[7; 2.88767513459]
Coordinates of the circumscribed circle: U[8; 1.15547005384]
Coordinates of the inscribed circle: I[6; 3.46441016151]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.7876789298° = 141°47'12″ = 0.66769463445 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 98.21332107017° = 98°12'48″ = 1.42774487579 rad

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How did we calculate this triangle?

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 10 ; ; b = 14 ; ; c = 16 ; ;

1. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 10+14+16 = 40 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40 }{ 2 } = 20 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20 * (20-10)(20-14)(20-16) } ; ; T = sqrt{ 4800 } = 69.28 ; ;

4. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 69.28 }{ 10 } = 13.86 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 69.28 }{ 14 } = 9.9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 69.28 }{ 16 } = 8.66 ; ;

5. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 14**2+16**2-10**2 }{ 2 * 14 * 16 } ) = 38° 12'48" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10**2+16**2-14**2 }{ 2 * 10 * 16 } ) = 60° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 10**2+14**2-16**2 }{ 2 * 10 * 14 } ) = 81° 47'12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 69.28 }{ 20 } = 3.46 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 38° 12'48" } = 8.08 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 14**2+2 * 16**2 - 10**2 } }{ 2 } = 14.177 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 16**2+2 * 10**2 - 14**2 } }{ 2 } = 11.358 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 14**2+2 * 10**2 - 16**2 } }{ 2 } = 9.165 ; ;
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